Physics
Dielectric Boundary Conditions
Dielectric boundary conditions refer to the rules that govern the behavior of electric fields at the interface between different dielectric materials. These conditions dictate how the electric field and its normal component change across the boundary. They are crucial for understanding the behavior of electric fields in materials and devices such as capacitors and waveguides.
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12 Key excerpts on "Dielectric Boundary Conditions"
eBook - PDFComputational Photonics
An Introduction with MATLAB
- Marek S. Wartak(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Further, for the dielectric medium the main role is played by ε. The medium is known as linear if ε is independent of E ; otherwise it is nonlinear. If it does not depend on position in space, the medium is said to be homogeneous; otherwise it is inhomogeneous. If properties are independent of direction, the medium is isotropic; otherwise it is anisotropic. 3.2 Boundary conditions Boundary conditions are derived from an integral form of Maxwell’s equations. For that purpose we separate all vectors into two components, one parallel to the interface and one normal to the interface. The derivation of boundary conditions is facilitated by using the contour and cylindrical shapes as shown in Fig. 3.1. It will be done independently for the electric and magnetic fields. 3.2.1 Electric boundary conditions First, analyse transversal components. Integrate Eq. (3.13) over closed loop C, see Fig. 3.1 and then set h → 0 ABCDA E · d l = −E 1 · d l + E 2 · d l = −E 1t w + E 2t w = 0 37 Boundary conditions A B C D Medium 2 Medium 1 n 1 n 2 E 2 D 2 D 1 E 1 ε r1 ε r2 C w h h ρ s Fig. 3.1 An interface between two media. Contour and volume used to derive boundary conditions for fields between two different dielectrics are shown. Therefore E 1t = E 2t (3.15) The tangential component of the electric field is therefore continuous across the boundary between any two dielectric media. Using general relation (3.6) one obtains the boundary condition for tangential component of vector D D 1t ε r1 = D 2t ε r2 (3.16) In order to obtain conditions for normal components, consider the cylinder shown in Fig. 3.1. Here, n 1 and n 2 are normal vectors pointing outwards of the top and bottom surfaces into corresponding dielectrics. Apply Gauss’s law with integration over surface S of the cylinder S D · d s = top D 1 · n 1 ds + bottom D 2 · n 2 ds = ρ s s The contribution from the outer surface of the cylinder vanishes in the limit h → 0.
eBook - PDFEngineering Electromagnetics
Pergamon Unified Engineering Series
- David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
The basic happening in dielectric materials is polarization, or separation of charge centers of orbital electrons and nuclei, of positive and negative ions with-out physically breaking their bonds and freeing them. EXERCISES 4-1. The plane z = 0 is the boundary between dielectric and free space. Near z = 0 the polarization, P, varies as shown in Fig. E.4-1. Estimate p P everywhere, including surface polarization charge, p SP , on the dielectric surface. 4-2. A solid dielectric cylinder of length L and radius a is uniformly polarized with Fig. E.4-1. Polarization near a boundary. 138 Dielectric Materials axially directed polarization, P. Determine the electric field along the axis, both within and outside the cylinder. 4-3. Show from the definition of P that dP/dt has dimensions of a current density, J P . Show }p is consistent with the expression, where p, v are charge density and velocity, respectively. 4-4. An electric field of 10,000 V/m is applied to a helium gas. For atom density n = 5 x 10 25 atoms/m 3 and an average electron cloud shift of 10 18 m find the polarization, P, polarization charge, p P , and permittivity of the helium. 4-5. A coaxial capacitor is partially filled with dielectric as shown. The inner and outer conductor radii are a, c respectively. The dielectric fills the coaxial to radius b. Find the fields. How does the presence of dielectric affect possible breakdown of the air? 4-6. A long coaxial capacitor is immersed in water (e=8l€ 0 ) (see Fig. E.4-6). If a voltage V 0 is applied, how far will the liquid rise in the capacitor? 4-7. The plasma frequency is the natural oscillatory frequency of an ionized gas. Fig.E.4-5. Dielectric loaded coaxial capacitor. Assuming 100% ionization, what is the plasma frequency of a hydrogen gas (density 10-3 mmHg)? HINT: The force on an electron in an electric field is eE, which causes motion predicted by Newton's Law, 4-8.
- Maurice Weiner(Author)
- 2010(Publication Date)
- World Scientific(Publisher)
In the following we discuss several kinds of boundary conditions one is likely to encounter in the TLM analysis. The two most common boundary conditions are the dielectric-dielectric and dielectric-conductor interfaces. It is difficult to imagine an electromagnetic configuration without the presence of one or the other of these two interfaces. How-ever other kinds of boundaries are possible. For simulation purposes the idealized dielectric-infinite permeability interface, or “open circuit”, provides a convenient means for obtaining total reflection of the wave energy from a given surface The open circuit is often useful since it may be used to approximate experimental conditions in which the radiated wave energy at a given surface is small, due to a very large mismatch in impedance levels. A conducting surface also will provide total reflection but with a resultant field inversion. 234 Electromagnetic Analysis Using Transmission Line Variables Other boundary conditions relate to the input/output of the electro-magnetic signal. Here we specify the conditions under which electro-magnetic energy is introduced into or leaves the region of interest, whether it be a closed device or an antenna. The difference between the input and output energies results in either the dissipation or storage of energy in the region of interest. In the following we describe ways in which to simulate the boundary conditions, using the TLM cell matrix and appropriate values of node resistors and transmission line imped-ances. Only symmetric variables are used but this does not alter the dis-cussion in any significant way. 5.1 Dielectric-Dielectric Interface There are several choices for positioning the TLM boundary of a di-electric-dielectric interface, two of which are shown in Fig.5.1. The side view shows the dielectric-dielectric interface with constants ε 1 and ε 2 . The permeability μ is assume to be the same in both regions.
eBook - PDF- M. Sibley(Author)
- 1995(Publication Date)
- Butterworth-Heinemann(Publisher)
6.3 Boundary relationships 157 much practical benefit. After all, we are quite happy to work with the relative permittivity of a material, so why should we concern ourselves with polariza-tion? The answer to this questionshould become clear in the next section when we examine whathappensto an electric field as it crosses a dielectric boundary. 6.3 BOUNDARY RELATIONSHIPS In the example at the end of the last section we encountereda composite dielectric capacitor. Thismeantthat the electric field in thecapacitorhad to cross aboundarybetween two dielectrics. The question of what happens to the E field is veryimportant-light and radio waves arelectromagnetic in form, and it would be nice to know what happens to these signals as they pass through a dielectric. Figure 6.5(a) shows an electric field crossing the boundarybetweentwo dielectrics. To analyse this situationwe will split the incident andtransmitted field into their tangential andperpendicularcomponents. Figure 6.5(b) shows the tangentialcomponentof the E field eitherside of the boundary. Let us consider a rectangularpath ABeD that lieseitherside of the boundary. Now, the work done in moving a unit point charge from point A around the path and back again must equal zero. (We have moved the charge around a closed loop, and so have not gained or lost potential.)So, £, (a) ;----..... Area ds (b) Closed path of length dl (c) Fig. 6.5 (a) An electric field crossing the boundary between two dielectrics; (b) tangential E field at the boundary; and (c) normal component of D field at the boundary 6,3 Boundary relationships 157 much practica l benefit After all, we are quite happy to work with the relative permittivity of a material , so why shoul d we concer n ourselves with polariza-tion? The answe r to this question shoul d become clear in the next section when .
eBook - PDFSteady Electric Fields and Currents
Elementary Electromagnetic Theory
- B. H. Chirgwin, C. Plumpton, C. W. Kilmister(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
However, for the steady conditions of an electrostatic field of moderate intensity, K may be taken to be independent of the field quantities. When an insulator is situated in a field which presumably penetrates into the material of the insulator we call the substance of the insulator a dielectric. We now modify our theory to take account of the presence of dielectrics; our choice of modification is suggested by the results described above and by our rough ideas of the structure of matter as outlined in Chapter 1. When a dielectric is situated in a field the positive and negative charges of 111 112 ELEMENTARY ELECTROMAGNETIC THEORY which the dielectric is constructed experience forces in opposite directions along the lines of force of the field. The characteristic of an insulator (in contrast to a conductor) is that in response to such forces these charges are displaced, or strained, from their normal equilibrium positions and take up new equilibrium positions. (In a conductor the charges move until any excess charge is situated on the surface of the conductor.) In these new positions of equilibrium a pair of positive and negative charges which originally neutra-lized each other are separated by a short distance, and so constitute a close approximation to a dipole. Thus, in addition to any simple charge which may be situated on or inside the dielectric, every element of volume becomes polarized. Before developing the analysis in the general case we consider the effect in the special uniform case of a parallel plate condenser whose plates are separ-ated by a distance /. We consider two situations: (1) The plates carry charges of density ±σ in the absence of a dielectric. Then D σ = ε 0 Ε, V = Et = σΐ/ε 0 , C = σ/Κ = ε 0 /ί, where V is the potential difference between the plates and C is the capacity of unit area of the (empty) condenser.
eBook - PDF- Jen-Shih Chang, Arnold J. Kelly, Joseph M. Crowley(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
4 Electrical Phenomena of Dielectric Materials R. Tobaz6on Centre National de la Recherche Scientifique Grenoble , France I. INTRODUCTION If we exclude metals, all remaining materials are dielectrics, whatever the state of the matter in question (solid, liquid, gas), and a permittivity e can be ascribed to any substance, with vacuum as the reference dielectric. Dielectrics can be employed either as passive devices (capacitors, ca bles) or in active devices (electrets, electrostatic motors), and they are required to function in our near or far environment (air, seawater, soil, space). Generally, materials are subjected to a voltage (dc, ac, impulse), and, in exceptional cases, to an electromagnetic field produced by, for example, an intense laser beam. The spatiotemporal distribution of the field inside the matter not only is imposed by the geometry of the elec trodes (whether to insure a uniform or a nonuniform field) and the shape of the voltage wave but also depends on space charges: charge carriers can be generated or blocked at interfaces or interphases, when different dielectric substances come into contact with each other. Among environmental constraints, we may consider the actions of pres sure, temperature, radiation, chemical attack, etc. Time is often a funda mental parameter in the study of dielectrics, e.g.: A perfect insulator would be a medium through which no conduction current could flow. In fact, the transition from “capacitive” behavior to “resistive” behavior depends on the conduction relaxation time 51 52 TOBAZEON tc = ep (insofar as a resistivity p can be ascribed to the material). Thus a dielectric may behave in a completely different manner under dc, ac, or impulse voltages.
eBook - PDF- Nikolai Bagdassarov(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
It refers to the phenomena of charge separation over a considerable distance in heterogeneous media at interfaces having contrasting dielectric constants ðε i Þ and specific electrical conductivities ðσ i Þ. The interfacial polarization in rocks operates at frequencies below ~1 GHz. In Figure 9.5 a contact between two dielectrics with contrasting conductivity σ 1 > σ 2 and dielectric constants ε 1 < ε 2 is shown, and in each medium the dielectric constant is frequency dependent: ε i ¼ ε i j σ i ω (i = 1, 2 and j ¼ ffiffiffiffiffiffi 1 p ). The boundary conditions at the interface between two dielectric media are as follows: (1) the tangential component of the electric field is continuous, E t 1 → ¼ E t 2 → , and (2) the normal component of the electric displacement field is continuous: D n 1 → ¼ D n 2 → ¼ ε 1 E n 1 → ¼ ε 2 E n 2 → . If the electric potential applied on external electrodes is V , then, due to E i → ¼ gradV , for the linear potential gradient one gets 1 2 E n 1 → þ 1 2 E n 1 → ¼ gradV (continuity of the electric field on the interface). Solving these two boundary conditions results in E n 1 → ¼ ε 2 ε 1 þ ε 2 ð2 gradV Þ and E n 2 → ¼ ε 1 ε 1 þ ε 2 ð2 gradV Þ. One assumes that the charge relaxation times of two dielectrics are τ 1 = ε 1 /σ 1 < τ 2 = ε 2 /σ 2 . At low frequencies ω < 1/τ 1 , in the dielectric medium 1 (on the left in Figure 9.5a) the electric charge is transported to the interface faster than the charge on the adjacent side is removed away from the interface. Finally, the electric charge accumulates at the interface, and the interface polarization is in favor of a more conductive material. At high frequencies ω > 1/τ 2 , the oscillations of the electric field are so fast that the electric charge transport into and out of the interface does not follow the external electric field, and in this case the charge accumulation on the boundary is in favor of a more polarized material.
eBook - PDF- Pradeep Fulay, Jung-Kun Lee(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
241 7 Linear Dielectric Materials 7.1 DIELECTRIC MATERIALS A dielectric material typically is a large-bandgap semiconductor ( E g ~ >4 eV) that exhibits high resistivity ( ρ ). The prefix dia means through in the Greek language. The word dielectric refers to a material that normally does not allow electricity (electrons, ions, and so on) to pass through it. There are special situations (for example, exposure to very high electric fields or changes in the composi-tion or microstructure) that may lead to a dielectric material exhibiting semiconducting or metallic behavior. However, when the term dielectric material is used, it generally is understood that the material essentially is a nonconductor of electricity. An electrical insulator is a dielectric material that exhibits a high breakdown field. 7.1.1 E LECTROSTATIC I NDUCTION To better understand the behavior of nonconducting materials, let us first examine the concept of electrostatic induction and what is meant by the terms free charge and bound charge . First, con-sider a dielectric such as a typical ceramic or a plastic that has a net positive charge on its surface. Now, assume that we bring a Conductor B near this charged Insulator A (Figure 7.1a). The electric field associated with the positively charged Insulator A pulls the electrons toward it from Conductor B. This is also described as the atoms in Conductor B being polarized or affected by the presence of an electric field. The process of the development of a negative charge on Conductor B is known as electrostatic induction. In this case, the negative charge developed on Conductor B is the bound charge because it is bound by the electric field caused by the presence of the charged Insulator A next to it. The creation of a bound negative charge on Conductor B, in turn, creates a net positive charge on the other side of the conductor because the conductor itself cannot have any net electric field within it.
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
(P3. 5 . 1 . 6 ) This equation is known as Gauss’s law for the electric displacement . In differential form, this law reads div D = ρ f . ◮ Isotropic dielectric. The polarization density P at a given point in a dielectric is determined by the external electric field E . For not too strong fields, P is linearly dependent on E . Furthermore, P bardbl E in an isotropic dielectric, and hence P = ε 0 χ E , (P3. 5 . 1 . 7 ) where χ is the electric susceptibility of the dielectric material. Then the electric displacement can be expressed as D = ε 0 E + P = ε 0 E + ε 0 χ E = ε 0 ε E , (P3. 5 . 1 . 8 ) where ε = 1 + χ is the electric permittivity of the material. ◮ Boundary conditions at dielectric interface. For two dielectric media meeting along a common interface, the following boundary conditions for tangential and normal components hold: E 1 τ = E 2 τ , P 2 n – P 1 n = – σ b , D 2 n – D 1 n = σ f . (P3. 5 . 1 . 9 ) The normal n points from medium 1 to medium 2. If there is no free charge at the interface, the normal components of the displacement are equal to each other. The first condition in (P3.5.1.9) results from the potentiality of the field E (the work along a closed path is zero). The second and third conditions follow from formulas (P3.5.1.2) and (P3.5.1.6), respectively. For an isotropic dielectric, we have E 1 τ = E 2 τ , ε 1 E 1 n = ε 2 E 2 n ; D 1 n = D 2 n , D 1 τ ε 1 = D 2 τ ε 2 . (P3. 5 . 1 . 10 ) It can be seen that the field lines are refracted at the interface between dielectric media. Note that unlike the electric field, the electric displacement lines are continuous at the interface. P3.5. E LECTROSTATICS OF D IELECTRICS 483 ◮ Calculation of fields in the presence of dielectrics. Given the distribution of free charges, the field in the whole space is uniquely determined by the equations for E and D together with the relation between them (P3.5.1.8) and the boundary conditions (P3.5.1.10). The solution has the following important properties: 1.
- Prabhakar H. Pathak, Robert J. Burkholder(Authors)
- 2021(Publication Date)
- Wiley-IEEE Press(Publisher)
CHAPTER 2 EM BOUNDARY AND RADIATION CONDITIONS 2.1 EM Field Behavior Across a Boundary Surface At macroscopic scales, the boundary surface between two different media is sharply defined. As a result, the macroscopic EM fields can exhibit a discontinuous behavior across such a boundary. Also, EM fields can become discontinuous across a boundary defined by some impressed surface source distribution of current, or charge, whether or not the medium on either side of the surface source distribution is dissimilar. Furthermore, it may so happen in the method of solution that different EM field expansions are chosen for representing the EM fields within different regions into which the problem configuration space is divided for convenience. In such instances, the EM fields have to be properly related across the artificial boundaries created by the analytical formulation of the solution. Therefore, it becomes necessary to develop boundary conditions that serve to relate the behavior of EM fields across boundary surfaces which may arise from the interface between two dissimilar media and/or surface sources, or from analytical formulations where different field representations are joined across mathematical surfaces. These boundary conditions are developed below. Consider a tiny, mathematical pillbox shape consisting of a right circular cylindrical surface of volume V with end caps as shown in Figure 2.1, where V is located with its center on the locally plane but otherwise smooth and curved boundary surface S b , which divides the surrounding space into regions 1 and 2 that, in general, can exhibit different electrical properties, respectively, on either side of S b .
- Rohit P. Prasankumar, Antoinette J. Taylor(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
The specific form of the boundary conditions shown above is valid in regions of space in which there are no charges or cur-rents ( ρ = J = 0). Furthermore, (1.21) through (1.24) allow us to explicitly state the relation-ships between tangential and normal components of the field at the interface. For example, (1.21) and (1.22) allow us to specify the relationship between the normal components of D − and B − as D D B B n n (2) n n and ( ) ( ) ( ) 1 1 2 = = (1.25) and (1.23) and (1.24) allow us to specify the relationship between the tangential components of E − and H − as E E H H t t t t and ( ) ( ) ( ) ( ) . 1 2 1 2 = = (1.26) where the subscripted n’s in (1.25) stand for normal components and the subscripted t’s in (1.26) stand for the tangential components. Thus we may summarize the boundary conditions by stating that the normal components of the electric displacement and the magnetic flux density are continuous at the interface, and that the tangential components of the electric field strength and the magnetic field strength are continuous at the interface. O P T I C A L T E C H N I Q U E S F O R S O L I D -S T A T E M A T E R I A L S C H A R A C T E R I Z A T I O N 8 1.1.3 Optical Constants The relation between the microscopic and macroscopic forms of Maxwell’s equations was first investigated by Lorentz (Choy 1999, Sihvola 1999) and is a subject detailed in vari-ous condensed matter texts (Ashcroft and Mermin 1976, Kittel 1996). However, an earlier connection between the atomic polarizability and the electric permittivity was developed by Mossotti in 1850, and independently by Clausius in 1879.
eBook - PDFInfrared and Millimeter Waves V8
Electromagnetic Waves in Matter, Part I
- Kenneth J. Button(Author)
- 1983(Publication Date)
- Academic Press(Publisher)
INFRARED AND MILLIMETER WAVES, VOL. 8 CHAPTER 1 Properties of Dielectric Materials G. W. Chantry National Physical Laboratory Teddington, Middlesex United Kingdom I. II. III. IV. V. INTRODUCTION THE MACROSCOPIC THEORY A. The Response Function B. The Correlation Function and the Cole-Cole Plot C. Corrections for the Internal Field D. Causality and the Kramers-Kronig Relations THE MICROSCOPIC THEORY EXPERIMENTAL METHODS SOME ILLUSTRATIVE EXAMPLES OF SUBMILLIMETER DIELECTRIC MEASUREMENTS A. Polar Liquids B. Nonpolar Liquids C. Solutions of Polar Molecules in Nonpolar Solvents D. Polymers E. Plastic Crystals F. Glasses REFERENCES 1 5 5 9 15 16 21 26 30 30 31 35 38 45 46 47 I. Introduction The study of liquids and polymers by means of far-infrared and submilli-meter spectroscopy forms part of the much larger topic of dielectric physics. A dielectric medium is one in which there are no free charges so the dc conductivity is zero, but the medium can sustain displacement currents and these may have lossy components. Thus dielectrics are all materials that are not metallic, semiconducting, or ionized. If an external field E is applied to a dielectric, the field inside the dielectric is given by D = (e/e 0 )E, (1) where ε is the permittivity of the dielectric and ε 0 the absolute permittivity of free space (8.85418 X 10 12 F/m). In nearly all dielectric work, however, it is customary to write ε = ε/ε 0 to avoid the constant apperance of the ε 0 factor; then ε so defined is the relative permittivity of the medium, i.e., the permit-1 Copyright © 1983 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-147708-8 2 G. W. CHANTRY tivity relative to that of the vacuum. The relative permittivity is complex because it will have a lossy component, and one therefore usually writes g = β ' -/ β , (2) where the caret is used to signify an explicitly complex quantity.
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